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# Simple Helices on Fano Threefolds

Published:2010-12-31
Printed: Sep 2011
• A. Polishchuk,
Department of Mathematics, University of Oregon, Eugene, OR 97405, U.S.A.
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## Abstract

Building on the work of Nogin, we prove that the braid group $B_4$ acts transitively on full exceptional collections of vector bundles on Fano threefolds with $b_2=1$ and $b_3=0$. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds with $b_2=1$ and very ample anticanonical class, every exceptional coherent sheaf is locally free.
 MSC Classifications: 14F05 - Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14J45 - Fano varieties

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